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In this text, we work in the Many Worlds interpretation of quantum mechanics.
In quantum mechanics, we should not let the wave function "collapse" at many places and occasions. A collapse cuts off information from the complete wave function of the universe. It is like a dramatic pruning of the network of Many Worlds, a pruning which will spoil the predictions of quantum mechanics.
The Einstein-Podolsky-Rosen experiment
A <-------- antiparallel spins ------> B
C
Let us consider the standard Einstein-Podolsky-Rosen thought experiment. Hidden variables which would specify the direction of the spins in the two particles would, in effect, mean that the wave function has collapsed immediately after the preparation of the system.
Bell's inequality shows that if we assume such a collapse, then we have to assume faster-than-light communication between A and B to make the results match the predictions of quantum mechanics.
A collapse reduces a complex-valued wave function to a real value. It is no wonder that with real values it is hard to reproduce the interference pattern of the original complex-valued wave functions.
Let A and B communicate their measurement results to a third observer C, who will make an interference pattern from the signals S_i he received from A and B. If we assume a collapse of the wave function at A and B, we cut off the necessary information which C needs to form the interference pattern. The interference pattern is very faint: the interference happens only between those signals S1, S2, for which the "state" s(A, B) after sending S1 is equal to the state s(A, B) after sending S2.
In practical setups, C would not observe any interference because A and B are macroscopic objects. The state s(A, B) will never occur again after sending a signal S_i.
The standard quantum mechanical measurement
quantum system
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measurement apparatus
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gauge
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C
The standard quantum mechanical experiment involves a physical system, a measurement apparatus, and one researcher C who looks at the gauge of the measurement apparatus and makes the wave function to collapse.
Wigner's friend
We may ask if having one observer C who makes the wave function to collapse is still too many? The network of the branches of the Many Worlds is cut when C looks at the gauge. Is it so that the branches we cut off are no longer needed in calculating the probability amplitudes?
quantum system
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measurement apparatus
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gauge
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C
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v
D
Let us have another human D. Let D form an interference pattern from the measurement results which C communicates to him. In standard quantum mechanics, the signals which D receives from C are complex-valued wave functions and D can form an interference pattern from them. If C would cause a wave function collapse when he looks at the gauge after a measurement C made, then there would be no interference pattern from the signals which D receives from C.
The argument above shows that NO observer can ever cause a collapse of a wave function. All the information in the complete network of the Many Worlds has to be preserved, to reproduce the predictions of standard quantum mechanics.
In the famous Schrödinger's cat thought experiment, the cat cannot make the wave function to collapse, because the experiment would then produce results which are not consistent with quantum mechanics. Since any scientist, including myself, can be "Schrödinger's cat" to another scientist, no one can make the wave function to collapse.
The view of a conscious subject
The above is at odds with our personal experience as conscious subjects: we definitely observe the gauge of an experiment to take a real value. We do not observe a complex-valued probability amplitude. How can we reconcile this with the fact that nothing is ever pruned from the network of Many Worlds?
In our thought experiment, let C come to D and ask to see the results of the experiment. D shows the list of individual measurement results.
In rare cases, the state of C, s(C) was identical after he sent signals S1 and S2. The signals S1 and S2 formed an interference pattern. C cannot reliably remember what he observed in S1 and S2. If he did, there would be no interference. In this thought experiment, the interference of different branches of Many Worlds creeps in because C does not have a 100% reliable memory.
How does a conscious subject end up in a certain branch of Many Worlds? David Bohm's model solves that in the nonrelativistic (the Schrödinger equation) case through hidden variables. The hidden variables would determine where the subject sails in the network of Many Worlds.
We are not sure if anyone has been able to solve the technical problems when quantum field theory is incorporated to Bohm's model. John Bell worried that Lorentz covariance cannot be satisfied in Bohm's model. If the number of particles varies, how can one specify hidden variables for them?
Conclusions
Bell's inequality is one demonstration of the fact that we cannot prune any parts of the Many Worlds network. No collapse is consistent with standard quantum mechanics. The Copenhagen interpretation works in practice, but theoretically it is inconsistent with quantum mechanics.
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